Let $S_\omega$ denote the group of permutations of the set $\omega$ of non-negative integers. The "wobbly group" $W$ is a normal subgroup of $S_\omega$ defined by $$W = \{f\in S_\omega: \sup\{|f(n)-n|:n\in \omega\} < \infty\}.$$ Is $S_\omega/W$ isomorphic to some subgroup of $S_\omega$?

Let $S_\omega$ denote the group of permutations of the set $\omega$ of non-negative integers. The "wobbling subgroup" $W$ is defined by $$W = \{f\in S_\omega: \sup\{|f(n)-n|:n\in \omega\} < \infty\}.$$ Is $S_\omega$ the only normal subgroup of $S_\omega$ containing $W$?